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Theorem sbcel2gv 2816
Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel2gv (B 𝑉 → ([B / x]A xA B))
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   𝑉(x)

Proof of Theorem sbcel2gv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . 2 (x = y → (A xA y))
2 eleq2 2098 . 2 (y = B → (A yA B))
31, 2sbcie2g 2790 1 (B 𝑉 → ([B / x]A xA B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  [wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  csbvarg  2871
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